An IntRoduction to grey methods by using R

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1 An IntRoduction to grey methods by using R Tan Xi Department of Statistics, NUFE Nov 6,

2 Contents: A brief introduction to Grey Methods An analysis of Degree of Grey Incidence Grey GM(, ) Model ()Construction (2)Test the accuracy of the model Some Envisions

3 A brief introduction to Grey Methods Who? Professor Deng! A Chinese! When? Long ago About 970s! What? White, Grey, Black? Application? Many realms! Economics, Physics, Social Science, and the list will go on!

4 An easy example Step by Step Suppose that the original sequence is Y { 0 = 8, 8.8,6,8, 24, 32 } comparative sequences are :. Y { = 0, 2.2, 9.28, 20.25, 23.4, } Y { 2 = 6, 6.35, 6.57, 6.98, 8.35, 8.75} Which of the comparative sequences is much closer to the original series? Y

5 Before computing the exact values, you can get the intuition by looking at the graph. (Ads: Where is Xie? ) GET THE INTUITION Y0 Y Y

6 Step: Initialize all sequences X X X 0 2 = {,.,2,2.25,3,4} = {,.22,.928, 2.205, 2.34, 3.069} = {,.0583,.0950,.633,.397,.4583} Step2 : Compute the absolute subtraction sequences Δ ( k) = Y ( k) Y( k) 0i 0 i

7 Step3: Compute the two-step minimum and maximum of the absolute subtraction sequences Δ = min min Y ( k) Y( k) = min 0 i k i Δ = max max Y ( k) Y( k) = 0 max 0 i k i

8 Step4: Compute coefficients of Grey incidence Formula: γ ( Y ( k), Y( k)) 0 i Δ + ρ Δ min max = Δ oi ( k) + ρ Δ max of which the distinguishing coefficient is 0.5. ρ

9 Step5: Compute the degree of Grey incidence 0 n γ( Y, Y) = γ( Y ( k), Yi( k)) 0 i 0 n k = γ ( Y, Y) = Y0 Y2 γ (, ) = And so our intuition is right! The computed results show that Y 0 is much more closer to Y than to Y 2, which is in coincidence with our intuition!

10 Recap: Initialize all sequences. Compute the absolute subtraction sequences Δ ( k) = Y ( k) Y( k) 0i 0 Compute the two-step minimum and maximum of the absolute subtraction sequences Δ = minmin Y ( k) Y( k) Δ max = maxmax Y0 ( k) Yi ( k) min 0 i k Compute coefficients of Grey incidence i Δ + ρ Δ γ ( Y ( k), Y( k)) = Δ + Δ min max 0 i oi ( k) ρ max i i k

11 Recap: Bingo!! Compute the degree of Grey incidence: n γ( Y, Y) = γ( Y ( k), Yi( k)) 0 i 0 n k =

12 All steps in one function Are you bored or puzzled with these steps?? Alternatives: The first: R functions! I ve involved all preceding steps in one function: 灰色关联分析函数.R I ll show you how to use it! The second: Click-Mouse Statistical Packages It s your choice! It s all up to you! For R-Users??

13 GM(, )Model GM(, ) type of Grey model is the most widely used in the literature, pronounced as Grey Model First Order One Variable. This model is a time series forecasting model. The differential equations of the GM(, ) model have time-varying coefficients

14 How to construct the GM(,) Model? Consider a time sequence X (0), which has n observations, X (0) = { X (0) (), X (0) (2),, X (0) ( n)} When this sequence is subjected to the Accumulating Generation Operation (AGO), the following sequence X () is obtained where () X = X () X () X () n { (), (2),, ( )} k () X k = X (0) i ( ) ( ) i=

15 How to construct the GM(,) Model? The grey difference equation of GM(,) is defined as follows: dx dt () () + = μ ax The least square estimate sequence of the grey difference equation of GM(,) is defined as follows: a μ ˆ α = = ( ) T T BB BY n

16 How to construct the GM(,) Model? Solve the grey difference equation of GM(,), the predicted GM(,) Model can be obtained: μ X k X e a ˆ () (0) ak ( + ) = () + To obtain the predicted value of the primitive data at time ( k+ H), the IAGO is used to establish the following grey model: (0) (0) b a( k+ H ) a X p ( k+ H) = [ X () ] e ( e ) a μ a

17 How to test the accuracy of the GM(,) Model? Residual Tests () i X () i Xˆ () i i,2,, n (0) (0) (0) Δ = = (0) Δ () i Φ ( i) = 00% i =, 2,, n (0) X () i

18 How to test the accuracy of the GM(,) Model? The Test of the degree of Grey incidence γ = n n ˆ (0) (0) ˆ (0) (0) ( X, X ) γ( X ( i), X ( i)) i= According to experience, the GM(,) Model is qualified if ˆ (0) (0) γ ( X, X ) > 0.6 ρ = 0.5, when

19 How to test the accuracy of the GM(,) Model? C and P Criteria S = [ X ( i) X ] (0) (0) 2 n S 2 = [ Δ ( i) Δ ] (0) (0) 2 n S S = 2 C= P = p Δ i Δ < S (0) (0) { ( ) }

20 How to test the accuracy of the GM(,) Model? C and P Criteria

21 An easy example executed by R Program Suppose the original sequence is: Construct the GM(,) Model and predict the values of 7~th Periods. R program : GM(,) 模型建立 检验和预测.R

22 Step: Construct the AGO sequence: Step2: Construct the matrix B and the vector Y n () () [ () (2)] X + X 2 () () [ (2) (3)] X + X () () B = [ X (3) + X (4)] = () () [ X (4) + X (5)] () () [ (5) (6)] X + X 2 (0) X (2) 3.5 (0) X (3) 32.8 = = (0) X (4) 34. (0) X (5) 35.8 (0) X (6) Y n

23 T Step3: Compute B B, ( T ) and BY. B B T n T B B = T ( BB) = T BY n = Step4: Solve the vector of parameters by using the least square estimate. ˆ α = a = μ =

24 Step5: Construct the GM(,) prediction Model dx dt dx dt () () () + =μ ax () X = ˆ () (0) μ ak μ X ( k+ ) = X () e + a a (0) μ X () = 26.7, = a So the GM(,) prediction model is: ˆ () k ( ) X k+ = e

25 Test the accuracy of the GM(,) Model Residual Test

26 Residual Test

27 The Test of the degree of Grey incidence γ X X = γ X i X i = n n ˆ (0) (0) ˆ (0) (0) (, ) ( ( ), ( )) i=

28 C and P Criteria S (0) (0) 2 [ X ( i) X ] = = n S 2 2 C= (0) (0) 2 [ Δ ( i) Δ ] = = n S S = 0 S = ei =Δ () i Δ (0) (0) = { , , , , , } P = p Δ i Δ < S = (0) (0) { ( ) }

29 GM(,) Model can be used to predict

30 R package? Some Envisions Any existing R package? Or can we write the first one? Collaboration More R programs and R Functions On grey methods?

31 Reference: [] 邓聚龙, 灰理论基础 [M]. 上海 : 华中科技大学出版社, [2] 刘思峰, 谢思明, 灰色系统理论及其应用 [M]. 北京 : 科学出版社,2008. [3] 王庚, 现代数学建模方法 [M]. 北京 : 科学出版社,2008. [4] 徐国祥, 统计预测与决策 [M]. 上海 : 上海财经大学出版社,2005. [5] Erdal Kayacan, Baris Ulutas, Okyay Kaynak, Grey system theory-based models in time series prediction[j]. Expert Systems with Applications, 200:

32 Acknowledgements I am grateful to all members of COS, without your excellent work, no R conferences could be held currently in China. I am also grateful to Professor Wang Gen for his help and suggestions on this topic. And thank everyone here for your patient listening and welcome any suggestions

33 Thank you! Contact Information: Tel:

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